simple path covers in graphs

International J.Math. Combin. Vol.3 (2008), 94-104

Simple Path Covers in Graphs

S. Arumugam and I. Sahul Hamid

National Centre for Advanced Research in Discrete Mathematics of

Kalasalingam University, Anand Nagar, Krishnankoil-626190, INDIA.

E-mail: s arumugam [email protected]

Abstract: A simple path cover of a graph G is a collection ψ of paths in G such that every

edge of G is in exactly one path in ψ and any two paths in ψ have at most one vertex in

common. More generally, for any integer k ≥ 1, a Smarandache path k-cover of a graph G

is a collection ψ of paths in G such that each edge of G is in at least one path of ψ and

two paths of ψ have at most k vertices in common. Thus if k = 1 and every edge of G is

in exactly one path in ψ, then a Smarandache path k-cover of G is a simple path cover of

G. The minimum cardinality of a simple path cover of G is called the simple path covering

number of G and is denoted by πs(G). In this paper we initiate a study of this parameter.

Key Words: Smarandache path k-cover, simple path cover, simple path covering number.

AMS(2000): 05C35, 05C38.

§1. Introduction

By a graph G = (V,E) we mean a finite, undirected graph with neither loops nor multiple edges.

The order and size of G are denoted by p and q respectively. For graph theoretic terminology

we refer to Harary [5]. All graphs in this paper are assumed to be connected and non-trivial.

If P = (v0, v1, v2, . . . , vn) is a path or a cycle in a graph G, then v1, v2, . . . , vn−1 are called

internal vertices of P and v0, vn are called external vertices of P . If P = (v0, v1, v2, . . . , vn) and

Q = (vn = w0, w1, w2, . . . , wm) are two paths in G, then the walk obtained by concatenating P

and Q at vn is denoted by P ◦Q and the path (vn, vn−1, . . . , v2, v1, v0) is denoted by P−1. For a

unicyclic graph G with cycle C, if w is a vertex of degree greater than 2 on C with deg w = k,

let e1, e2, . . . , ek−2 be the edges of E(G) −E(C) incident with w. Let Ti, 1 ≤ i ≤ k − 2, be the

maximal subtree of G such that Ti contains the edge ei and w is a pendant vertex of Ti. Then

T1, T2, . . . , Tk−2 are called the branches of G at w. Also the maximal subtree T of G such that

V (T ) ∩ V (C) = {w} is called the subtree rooted at w.

The concept of path cover and path covering number of a graph was introduced by Harary

[6]. Preliminary results on this parameter were obtained by Harary and Schwenk [7], Peroche

[9] and Stanton et al. [10], [11].

1Supported by NBHM Project 48/2/2004/R&D-II/7372.2Received May 16, 2008. Accepted September 16, 2008.

Simple Path Covers in Graphs 95

Definition 1.1([6]) A path cover of a graph G is a collection ψ of paths in G such that every

edge of G is in exactly one path in ψ. The minimum cardinality of a path cover of G is called

the path covering number of G and is denoted by π(G) or simply π.

Theorem 1.2([10]) For any tree T with k vertices of odd degree, π(T ) = k2 .

Theorem 1.3([7]) The path covering number of the complete graph Kp is given by π(Kp) =⌈p2

⌉.

(For any real number x, ⌈x⌉ denotes the least positive integer ≥ x.)

Theorem 1.4([4]) Let G be a unicyclic graph with unique cycle C. Let m denote the number

of vertices of degree greater than 2 on C. Let k denote the number of vertices of odd degree.

Then

π(G) =

2 if m = 0

k2 + 1 if m = 1

k2 otherwise

Theorem 1.5([4]) For any graph G, π(G) ≥⌈

∆2

⌉.

The concepts of graphoidal cover and acyclic graphoidal cover were introduced by Acharya

et al. [1] and Arumugam et al. [4].

Definition 1.6([1]) A graphoidal cover of a graph G is a collection ψ of (not necessarily open)

paths in G satisfying the following conditions.

(i) Every path in ψ has at least two vertices.

(ii) Every vertex of G is an internal vertex of at most one path in ψ.

(iii)Every edge of G is in exactly one path in ψ.

If further no member of ψ is a cycle in G, then ψ is called an acyclic graphoidal cover of G.

The minimum cardinality of a graphoidal cover of G is called the graphoidal covering number of

G and is denoted by η(G). Similarly we define the acyclic graphoidal covering number ηa(G).

An elaborate review of results in graphoidal covers with several interesting applications

and a large collection of unsolved problems is given in Arumugam et al.[2].

For any graph G = (V,E), ψ = E is trivially an acyclic graphoidal cover and has the

interesting property that any two paths in ψ have at most one vertex in common. Motivated

by this observation we introduced the concept of simple acyclic graphoidal covers in graphs [3].

Definition 1.7([3]) A simple acyclic graphoidal cover of a graph G is an acyclic graphoidal

cover ψ of G such that any two paths in ψ have at most one vertex in common. The minimum

cardinality of a simple acyclic graphoidal cover of G is called the simple acyclic graphoidal

covering number of G and is denoted by ηas(G) or simply ηas.

Definition 1.8 Let ψ be a collection of internally disjoint paths in G. A vertex of G is said to

be an interior vertex of ψ if it is an internal vertex of some path in ψ, otherwise it is said to

96 S. Arumugam and I. Sahul Hamid

be an exterior vertex of ψ.

Theorem 1.9([3]) For any simple acyclic graphoidal cover ψ of a graph G, let tψ denote the

number of exterior vertices of ψ. Let t = min tψ, where the minimum is taken over all simple

acyclic graphoidal covers ψ of G. Then ηas(G) = q − p+ t.

Theorem 1.10([3]) Let G be a unicyclic graph with n pendant vertices. Let C be the unique

cycle in G and let m denote the number of vertices of degree greater than 2 on C. Then

ηas(G) =

3 if m = 0

n+ 2 if m = 1

n+ 1 if m = 2

n if m ≥ 3

Theorem 1.11([3]) Let m and n be integers with n ≥ m ≥ 4. Then

ηas(Km,n) =

mn−m− n if n ≤(m2

)

mn−m− n+ r if n =(m2

)+ r, r > 0.

In this paper we introduce the concept of simple path cover and simple path covering

number πs of a graph G and initiate a study of this parameter. We observe that the concept

of simple path cover is a special case of Smarandache path k-cover [8]. For any integer k ≥ 1,

a Smarandache path k-cover of a graph G is a collection ψ of paths in G such that each edge

of G is in at least one path of ψ and two paths of ψ have at most k vertices in common. Thus

if k = 1 and every edge of G is in exactly one path in ψ, then a Smarandache path k-cover of

G is a simple path cover of G.

§2. Main results

Definition 2.1 A simple path cover of a graph G is a path cover ψ of G such that any two

paths in ψ have at most one vertex in common. The minimum cardinality of a simple path

cover of G is called the simple path covering number of G and is denoted by πs(G). Any simple

path cover ψ of G for which |ψ| = πs(G) is called a minimum simple path cover of G.

Example 2.2 Consider the graph G given in Fig.2.1.


v1 v2 v3

v4

v6 v5 v7 v8

Fig. 2.1

Then ψ = {(v1, v4, v7, v8), (v3, v4, v5, v6), (v2, v4), (v7, v5)} is a minimum simple path cover of G

so that πs(G) = 4.

Remark 2.3 Every path in a simple path cover of a graph G is an induced path.

Theorem 2.4 For any simple path cover ψ of a graph G, let tψ =∑P∈ψ

t(P ), where t(P ) denotes

the number of internal vertices of P and let t = max tψ, where the maximum is taken over all

simple path covers ψ of G. Then πs(G) = q − t.

Proof Let ψ be any simple path cover of G. Then

q =∑P∈ψ

|E(P )|

=∑P∈ψ

(t(P ) + 1)

= |ψ| + ∑P∈ψ

t(P )

= |ψ| + tψ

Hence |ψ| = q − tψ so that πs(G) = q − t. �

Corollary 2.5 For any graph G with k vertices of odd degree πs(G) = k2 +

∑v∈V (G)

⌊deg v

2

⌋− t.

Proof Since q = k2 +

∑v∈V (G)

⌊deg v

2

⌋the result follows. �

Corollary 2.6 For any graph G, πs(G) ≥ k2 where k is the number of vertices of odd degree in

G. Further, the following are equivalent.

(i) πs(G) = k2 .

(ii) There exists a simple path cover ψ of G such that every vertex v in G is an internal

vertex of⌊deg v

2

⌋paths in ψ.

(ii)There exists a simple path cover ψ of G such that every vertex of odd degree is an


external vertex of exactly one path in ψ and no vertex of even degree is an external vertex of

any path in ψ.

Remark 2.7 For any (p, q)-graph G, πs(G) ≤ q. Further, equality holds if and only if G is

complete. Hence it follows from Theorem 1.3 that πs(Kn) = π(Kn) if and only if n = 2.

Remark 2.8 Since any path cover of a tree T is a simple path cover of T , it follows from

Theorem 1.2 that πs(T ) = π(T ) = k2 , where k is the number of vertices of odd degree in T .

We now proceed to determine the value of πs for unicyclic graphs and wheels.

Theorem 2.9 Let G be a unicyclic graph with cycle C. Let m denote the number of vertices

of degree greater than 2 on C. Let k be the number of vertices of odd degree. Then

πs(G) =

3 if m = 0

k2 + 2 if m = 1

k2 + 1 if m = 2

k2 if m ≥ 3

Proof Let C = (v1, v2, . . . , vr, v1).

Case 1. m = 0.

Then G = C so that πs(G) = 3.

Case 2. m = 1.

Let v1 be the unique vertex of degree greater than 2 on C. Let G1 be the tree rooted at

v1. Then G1 has k vertices of odd degree and hence πs(G1) = k2 . Let ψ1 be a minimum simple

path cover of G1.

If deg v1 is odd, then degG1 v1 is odd. Let P be the path in ψ1 having v1 as a terminal

vertex. Now, let

P1 = P ◦ (v1, v2)

P2 = (v2, v3, . . . , vr) and

P3 = (vr , v1).

If deg v1 is even, then degG1 v1 is even. Let P = (x1, x2, . . . , xr, v1, xr+1, . . . , xs) be a path

in ψ1 having v1 as an internal vertex. Now, let

P1 = (x1, x2, . . . , xr , v1, v2)

P2 = (xs, xs−1, . . . , xr+1, v1, vr) and

P3 = (v2, v3, . . . , vr).

Then ψ = {ψ1 − {P}} ∪ {P1, P2, P3} is a simple path cover of G and hence πs(G) ≤|ψ1| + 2 = k

2 + 2. Further, for any simple path cover ψ of G, all the k vertices of odd degree

and at least two vertices on C are terminal vertices of paths in ψ. Hence t ≤ q− k2 − 2, so that

πs(G) = q − t ≥ k2 + 2. Thus πs(G) = k

2 + 2.


Case 3. m = 2.

Let v1 and vi, where 2 ≤ i ≤ r, be the vertices of degree greater than 2 on C. Let P and

Q denote respectively the (v1, vi)-section and (vi, v1)-section of C. Let vj be an internal vertex

of P (say). Let R1 and R2 be the (v1, vj)-section of P and (vj , vi)-section P respectively. Let

G1 be the graph obtained by deleting all the internal vertices of P .

Subcase 3.1 Both deg v1 and deg vi are odd.

Then both degG1 v1 and degG1 vi are even. Hence G1 is a tree with k − 2 odd vertices so

that πs(G1) = k2 − 1. Let ψ1 be a minimum simple path cover of G1 . Then ψ = ψ1 ∪ {R1, R2}

is a simple path cover of G and |ψ| = k2 + 1.Hence πs(G) ≤ k

2 + 1.

Subcase 3.2 Both deg v1 and deg vi are even.

Then degG1 v1 and degG1 vi are odd. Hence G1 is a tree with k+ 2 vertices of odd degree

so that πs(G1) = k2 + 1. Let ψ1 be a minimum simple path cover of G1.

Suppose v1 and vi are terminal vertices of two different paths in ψ1, say P1 and P2 respec-

tively. Now, let

Q1 = P1 ◦R1

Q2 = P2 ◦R−12 and

ψ = {ψ1 − {P1, P2}} ∪ {Q1, Q2}.Suppose there exists a path P1 in ψ1 having both v1 and vi as its end vertices. Then let

P1 = Q. Let P2 be an u1-w1 path in ψ1 having v1 as an internal vertex and P3 be an u2-w2 path

in ψ1 having vi as an internal vertex. Let S1 and S2 be the (u1, v1)-section of P2 and (w1, v1)-

section of P2 respectively. Let S3 and S4 be the (u2, vi)-section of P3 and (w2, vi)-section of P3

respectively. Now, let

Q1 = S1 ◦ P1 ◦ S−13

Q2 = S2 ◦R1

Q3 = S4 ◦R−12 and

ψ = {ψ1 − {P1, P2, P3}} ∪ {Q1, Q2, Q3}.Then ψ is a simple path cover of G and |ψ| = |ψ1| = k

2 + 1 and hence πs(G) ≤ k2 + 1.

Subcase 3.3 deg v1 is odd and deg vi is even.

Then degG1 v1 is even and degG1 vi is odd. Hence G1 is a tree with k vertices of odd degree

so that πs(G1) = k2 . Let ψ1 be a minimum simple path cover of G1. Let P1 be the path in ψ1

having vi as a terminal vertex.

If E(P1) ∩ E(Q) = φ, let

Q1 = P1 ◦R−12

Q2 = R1 and

ψ = {ψ1 − {P1}} ∪ {Q1, Q2}.Suppose E(P1) ∩ E(Q) 6= φ. Since degG1 vi ≥ 3, there exists an u1-w1 path in ψ1, say P2,

having vi as an internal vertex. Let S1 and S2 be the (w1, vi)-section of P2 and (u1, vi)-section

of P2 respectively. Now, let

Q1 = P1 ◦ S−11

Q2 = S2 ◦R−12


Q3 = R1 and

ψ = {ψ1 − {P1, P2}} ∪ {Q1, Q2, Q3}.Then ψ is a simple path cover of G and |ψ| = |ψ1| + 1 = k

2 + 1. Hence πs(G) ≤ k2 + 1.

Thus in either of the above subcases, we have πs(G) ≤ k2 + 1. Also, for any simple path

cover ψ of G all the k vertices of odd degree and at least one vertex on C are terminal vertices

of paths in ψ. Hence t ≤ q − k2 − 1, so that πs(G) = q − t ≥ k

2 + 1.

Hence πs(G) = k2 + 1.

Case 4. m ≥ 3.

Let vi1 , vi2 , . . . , vis , where 1 ≤ i1 < i2 < · · · < is ≤ r and s ≥ 3, be the vertices of degree

greater than 2 on C. Let ψij , 1 ≤ j ≤ s, be a minimum simple path cover of the tree rooted

at vij . Consider the vertices vi1 , vi2 and vi3 . For each j, where 1 ≤ j ≤ 3, let Pj be the

path in ψij in which vij is a terminal vertex if deg vij is odd, otherwise let Pj be an uj-wj

path in ψij in which vij is an internal vertex and Rj and Sj be the (uj, vij ) and (wj , vij )-

sections of Pj respectively. Further, let P = (vi1 , vi1+1, . . . , vi2 ), Q = (vi2 , vi2+1, . . . , vi3) and

R = (vi3 , vi3+1, . . . , vi1).

If deg vi1 , deg vi2 and deg vi3 are even, let Q1 = R1 ◦ P ◦ R−12 , Q2 = S2 ◦ Q ◦ R−1

3 and

Q3 = S3 ◦R ◦ S−11 .

If deg vi1 , deg vi2 and deg vi3 are odd, let Q1 = P1 ◦ P , Q2 = P2 ◦Q and Q3 = P3 ◦R.

If deg vi1 , deg vi2 are odd and deg vi3 is even, let Q1 = P1 ◦ P ◦ P−12 , Q2 = R3 ◦Q−1 and

Q3 = S3 ◦R.

If deg vi1 , deg vi2 are even and deg vi3 is odd, let Q1 = R1 ◦ P ◦ R−12 , Q2 = S2 ◦Q ◦ P−1

3

and Q3 = R ◦ S−11 .

Then ψ = (s⋃j=1

ψij − {P1, P2, P3}) ∪ {Q1, Q2, Q3} is a simple path cover of G such that

every vertex of odd degree is an external vertex of exactly one path in ψ and no vertex of even

degree is an external vertex of any path in ψ. Hence πs(G) = k2 . �

Corollary 2.10 Let G be as in Theorem 2.9. Then πs(G) = π(G) if and only if m ≥ 3.

Proof The proof follows from Theorem 2.9 and Theorem 1.4. �

We observe that there are infinite families of graphs such as trees and unicyclic graphs

having at least three vertices of degree greater than 2 on C for which πs = π and so the

following problem arises naturally.

Problem 2.11 Characterize graphs for which πs = π.

Theorem 2.12 For a wheel Wn = K1 + Cn−1, we have

πs(Wn) =

6 if n = 4⌊n2

⌋+ 3 if n ≥ 5

Proof Let V (Wn) = {v0, v1, . . . , vn−1} and E(Wn) = {v0vi : 1 ≤ i ≤ n− 1}∪ {vivi+1 : 1 ≤i ≤ n− 2} ∪ {v1vn−1}.


If n = 4, then Wn = K4 and hence πs(Wn) = 6.

Now, suppose n ≥ 5. Let r =⌊n2

⌋

If n is odd, let

Pi = (vi, v0, vr+i), i = 1, 2, . . . , r.

Pr+1 = (v1, v2, . . . , vr),

Pr+2 = (v1, v2r, v2r−1, . . . , vr+2) and

Pr+3 = (vr, vr+1, vr+2).

If n is even, let

Pi = (vi, v0, vr−1+i), i = 1, 2, . . . , r − 1 .

Pr = (v0, v2r−1),

Pr+1 = (v1, v2, . . . , vr−1),

Pr+2 = (v1, v2r−1, . . . , vr+1) and

Pr+3 = (vr−1, vr, vr+1).

Then ψ = {P1, P2, . . . , Pr+3} is a simple path cover ofWn. Hence πs(Wn) ≤ r+3 =⌊n2

⌋+3.

Further, for any simple path cover ψ of Wn at least three vertices on C = (v1, v2, . . . , vn−1) are

terminal vertices of paths in ψ. Hence t ≤ q− k2 − 3, so that πs(Wn) = q− t ≥ k

2 +3 =⌊n2

⌋+3.

Thus πs(Wn) =⌊n2

⌋+ 3. �

Remark 2.13 Since every simple acyclic graphoidal cover of a graph G is a simple path cover

of G and every simple path cover of G is a path cover of G, we have ηas ≥ πs ≥ π. These

parameters may be either equal or all distinct as shown below. For the graph G1 given in

Figure 2, ηas(G1) = 7, πs(G1) = 6, π(G1) = 5 and for the graph G2 given in Fig.2.2, we have

ηas(G2) = πs(G2) = π(G2) = 3.

G1G2

Fig.2.2

Problem 2.14 Characterize graphs for which ηas = πs = π.

We now proceed to obtain some bounds for πs.

Theorem 2.15 For any graph G, πs(G) ≥⌈

∆2

⌉. Further, the following are equivalent.

(i) πs(G) =⌈

∆2

⌉.

(ii) ηas(G) = ∆ − 1.

(iii) G is homeomorphic to a star.


Proof Since πs ≥ π, the inequality follows from Theorem 1.5.

Suppose πs(G) =⌈

∆2

⌉. Let ψ = {P1, P2, . . . , Pr}, where r =

⌈∆2

⌉be a minimum simple

path cover of G. Let v be a vertex of G with deg v = ∆. Then v lies on each Pi and

v is an internal vertex of all the paths in ψ except possibly for at most one path. Hence

V (Pi) ∩ V (Pj) = {v}, for all i 6= j, so that G is homeomorphic to a star. Obviously, if G is

homeomorphic to a star, then πs(G) =⌈

∆2

⌉. Thus (i) and (iii) are equivalent. Similarly the

equivalence of (ii) and (iii) can be proved. �

Theorem 2.16 For any graph G, πs(G) ≥(ω2

), where ω is the clique number of G.

Proof Let H be a maximum clique in G so that |E(H)| =(ω2

). Let ψ be a simple path

cover of G. Since any path in ψ covers at most one edge of H , it follows that πs(G) ≥(ω2

). �

In the following theorem we characterize cubic graphs for which πs =(ω2

).

Theorem 2.17 Let G be a cubic graph. Then πs(G) =(ω2

)if and only if G = K4.

Proof Let G be a cubic graph with πs(G) =(ω2

). Clearly ω = 3 or 4. Suppose ω = 3.

Then it follows from Corollary 2.6 that πs(G) ≥ p2 so that p = 6. Hence G is isomorphic to the

cartesian product K3 ×K2 and it can be shown that πs(K3 ×K2) = 6 6=(ω2

). Thus ω = 4 and

consequently G = K4. �

Problem 2.18 Characterize graphs for which πs(G) =(ω2

).

If ∆ ≤ 3, then every simple path cover of G is a simple acyclic graphoidal cover of G and

hence ηas(G) = πs(G). However, the converse is not true. For the complete graph Kp(p ≥ 5),

πs = ηas whereas ∆ ≥ 4. We now prove that the converse is true for trees and unicyclic graphs.

Theorem 2.19 Let G be a tree. Then ηas(G) = πs(G) if and only if ∆ ≤ 3.

Proof Let G be a tree with ηas(G) = πs(G).

Suppose ∆ ≥ 4. Let v be a vertex of G with deg v ≥ 4.

Let ψ be a minimum simple acyclic graphoidal cover of G. Let P1 and P2 be two paths in

ψ having v as a terminal vertex. Let Q = P1 ◦ P−12 . Since G is a tree, Q is an induced path

and hence ψ1 = (ψ − {P1, P2}) ∪ {Q} is a simple path cover of G with |ψ1| = |ψ| − 1 = ηas − 1

so that πs(G) ≤ ηas(G) − 1, which is a contradiction. Hence ∆ ≤ 3. �

Theorem 2.20 Let G be a unicyclic graph. Then ηas(G) = πs(G) if and only if ∆ ≤ 3.

Proof Let G be a unicyclic graph with ηas(G) = πs(G). Let k denote the number of

vertices of odd degree and n be the number of pendant vertices of G.

It follows from Theorem 1.10 and Theorem 2.9 that k = 2n. Now, suppose ∆ > 3. Then

2q =∑

v∈V (G)

deg v=1

deg v +∑

v∈V (G)

deg v>1

and is odd

deg v +∑

v∈V (G)

deg v>1

and is even

deg v

> n+ 3(k − n) + 2(p− k)

= 2p,


which is a contradiction. Hence ∆ ≤ 3. �

The above results lead to the following problem.

Problem 2.21 Characterize graphs for which ηas(G) = πs(G).

In the following theorem we establish a relation connecting the parameters ηas and πs.

Theorem 2.22 For any (p, q)-graph G, ηas(G) ≤ πs(G)+q−p+n− k2 , where n and k respectively

denote the number of pendant vertices and the number of odd vertices of G. Further, equality

holds if and only if πs(G) = k2 .

Proof Let ψ be a minimum simple path cover of G. Let i(v) denote the number of paths in

ψ having v ∈ V as an internal vertex. If i(v) ≥ 2, let Pi, where 1 ≤ i ≤ i(v), be the ui-wi path

in ψ having v as an internal vertex and let Qi and Ri, where 2 ≤ i ≤ i(v), respectively denote

the (v, wi)-section and (v, ui)-section of Pi. Let ψ1 be the collection of paths obtained from ψ

by replacing P2, P3, . . . , Pi(v) by Q2, Q3, . . . , Qi(v) and R2, R3, . . . , Ri(v) for every v ∈ V with

i(v) ≥ 2. Then ψ1 is a simple acyclic graphoidal cover of G with |ψ1| = πs(G)+∑

v∈V

i(v)≥2

(i(v)−1).

Since i(v) ≤⌊deg v

2

⌋, it follows that

ηas(G) ≤ πs(G) +∑

v∈V

deg v≥4

(⌊deg v

2

⌋− 1)

= πs(G) +∑

v∈V

deg v≥2

(⌊deg v

2

⌋− 1)

= πs(G) +∑

v∈V

deg v≥2

⌊deg v

2

⌋− (p− n)

= πs(G) +∑

v∈V

deg v≥2

and is odd

deg v−12 +

∑v∈V

deg v≥2

and is even

deg v2 − p+ n

= πs(G) +∑

v∈V

deg v≥2

and is odd

deg v2 − k−n

2 +∑

v∈V

deg v≥2

and is even

deg v2 − p+ n

= πs(G) +∑

v∈V

deg v≥2

deg v2 − k

2 + n2 − p+ n

= πs(G) +∑v∈V

deg v2 − k

2 − p+ n.

= πs(G) + q − p+ n− k2 .

Thus ηas(G) ≤ πs(G)+q−p+n− k2 . Further, it is clear that ηas(G) = πs(G)+q−p+n− k

2

if and only if there exist a minimum simple path cover ψ of G such that i(v) =⌊deg v

2

⌋for all

v ∈ V . Hence it follows from Corollary 2.6 that ηas(G) = πs(G) + q − p+ n− k2 if and only if

πs = k2 . �

Corollary 2.23 If πs(G) = k2 , then ηas(G) = q − p+ n.


Proof Suppose πs(G) = k2 . By Theorem 2.22, we have ηas(G) ≤ q−p+n. Hence it follows

from Theorem 1.9 that ηas(G) = q − p+ n. �

Remark 2.24 The converse of Corollary 2.23 is not true. For example, ηas(K4,5) = q−p = 11,

whereas πs(K4,5) > 2 = k2 .

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simple path covers in graphs

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